DOCUMENT RESUME ED 404 348 TM 026 105 AUTHOR Timm, Neil H. TITLE Full Rank Multivariate Repeated Measurement Designs and Extended Linear Hypotheses. PUB DATE Aug 96 NOTE 28p.; Paper presented at the Annual Meeting of the American Psychological Association (104th, Toronto, Canada, August 1996). PUB TYPE Reports Evaluative/Feasibility (142) Speeches /Conference Papers (150) EDRS PRICE MF01/PCO2 Plus Postage. DESCRIPTORS Equations (Mathematics); *Mathematical Models; *Multivariate Analysis IDENTIFIERS *Full Rank Linear Model; *Repeated Measures Design; Statistical Analysis System ABSTRACT Hypotheses that do not have the standard bilinear form theta=CBM=0 occur naturally in the analysis of repeated measurement designs. An expanded class of the tests of the form psi=Tr(G theta)=0, called extended linear hypotheses, provides a richer class of parametric functions. A method to analyze double multivariate and mixed multivariate models (MMM) using the Statistical Analysis System (SAS) is demonstrated. The analysis is extended to extended hypotheses, and a new approximate test of extended linear hypotheses for MMM designs is developed that does not require multivariate sphericity, but only a general Kronecker structure. Two appendixes provide SAS programs for these methods. (Contains 4 tables and 26 references.) (SLD) *********************************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. ***********************************************************************
29
Embed
DOCUMENT RESUME TM 026 105 AUTHOR Timm, Neil H. … · is the data matrix, XNxk is the full rank design matrix, rank (X) = k, Bkxpq is the matrix of fixed unknown location parameters,
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
DOCUMENT RESUME
ED 404 348 TM 026 105
AUTHOR Timm, Neil H.TITLE Full Rank Multivariate Repeated Measurement Designs
and Extended Linear Hypotheses.PUB DATE Aug 96NOTE 28p.; Paper presented at the Annual Meeting of the
American Psychological Association (104th, Toronto,Canada, August 1996).
PUB TYPE Reports Evaluative/Feasibility (142)Speeches /Conference Papers (150)
EDRS PRICE MF01/PCO2 Plus Postage.DESCRIPTORS Equations (Mathematics); *Mathematical Models;
*Multivariate AnalysisIDENTIFIERS *Full Rank Linear Model; *Repeated Measures Design;
Statistical Analysis System
ABSTRACTHypotheses that do not have the standard bilinear
form theta=CBM=0 occur naturally in the analysis of repeatedmeasurement designs. An expanded class of the tests of the formpsi=Tr(G theta)=0, called extended linear hypotheses, provides aricher class of parametric functions. A method to analyze doublemultivariate and mixed multivariate models (MMM) using theStatistical Analysis System (SAS) is demonstrated. The analysis isextended to extended hypotheses, and a new approximate test ofextended linear hypotheses for MMM designs is developed that does notrequire multivariate sphericity, but only a general Kroneckerstructure. Two appendixes provide SAS programs for these methods.(Contains 4 tables and 26 references.) (SLD)
Group x Cond 0.1907 (2, 0, 14) 0.9218 (6, 58) 0.4862
That the multivariate mixed model results may be obtained from the DMM, observe that for the test of
conditions given parallelism that the hypothesis test matrix
H* =
152.93 (Sym)
95.79 62.73
14.75 856 159
is the average of H11 and H22 where
'148.02 Sym) 4.89 Sym
H11 = 96.32 62.67 H22 = -053 0.06
13.38 8.71 1.21 1.36 -0.15 0.39
are the submatrices of H for the test of Hc given parallelism. The degrees of freedom for the mixed model analyses
are v: = vhu = vh rank (M) / p = vh rank(A) = 1.2 = 2 and v: = ye rank(M) / p = ye rank(A) = 16.6 / 3 = 32. The
results for the test of cond x group interaction follow similarly, Timm (1980) and Boik (1988). From the MMM
results, one can also obtain the univariate split-plot univariate F-ratios, a variable at a time.
When multivariate sphericity is not satisfied, one cannot perform a MMM analysis. Boik (1988, 1991)
developed an c- adjustment for the tests of conditions and conditions x group interaction following the work of Box.
Because the multivariate tests may be approximated by a Wishart distribution when the null hypothesis is true:
Es w(v: = Euve,Q,0)
11` Wp(v: = euvh,S2-IA,0)
Boik showed how one may use an estimate of 6 to adjust the approximate F tests for the multivariate criteria where
TrR Sii )21+ [Tr(ii1
Si=1 =
e = (4.17)
[Tr(Sy)]2 + TOO}a=1 j=1
u = rank(A) and vh and ve are the hypothesis and error degrees of freedom for the DMM tests. Boik (1991)
proposed an alternative adjustment factor that improves the approximation when SI has the general Kronecker
structure given in (2.22).
Using Wilks's A criterion, Boik (1988) shows how one may make the e- adjustment to the associated F-
statistic. The result applies equally to the other criteria using Rao's F-approximation. In Program 1.sas, we use To2,
the Bartlett-Lawley-Hotelling trace criterion. Then
F = 2(sN +1)7'02 I s2 (2 M + s + 1)(4.18)
F[s(2 M + s+ 1),2(sN + 1)]
where
s = min (vh,p), vh = euv v h = tuvh
M = (14 - p1-1)/ 2, N = (4 -p-1)/2.
From the program output, e = 0.7305055 and the e- adjusted MANOVA Table 3 for the MMM tests results.
12
14
Table 3
T02, e adjusted Analysis, Zullo Data
Hypothesis F df p-value
Cond/Parall 65.090 (4.38, 30.31) < 0.0001
Group x Cond 0.903 (4.38, 30.31) 0.4830
The p-value for the unadjusted MMM was 0.4862 for the test of parallelism. The comparison using Wilks's A
criterion is 0.463 versus 0.4687, for the adjusted MMM and the unadjusted test, respectively. The nominal p-values
for the two procedures are approximately equal, 0.483 vs. 0.463, for To2 and A, respectively, for the test of
parallelism.
It appears that we have three competing strategies for the analysis of a DMM. The DMM, the MMM, and the
8 adjusted MMM. Given multivariate sphericity, the MMM is of course most powerful. When multivariate
sphericity does not hold, neither the adjusted MMM or the DMM analysis is most powerful. Boik (1991)
recommends using the DMM. The choice between the two procedures depends on the ratio of the traces of the
noncentrality matrices of the associated Wishart distributions which is seldom known in practice.
Extended Linear Hypotheses - Example
Krishnaiah et al. (1980) developed a "Roots" program to test extended linear hypotheses using three test criteria:
Tmax2 , Roy's largest root and 7;2. They consider hypotheses using only a single matrix G. We illustrate their
procedure using Roy's largest root test and 7;2 using PROC IML and data from Timm (1975, p. 454) shown in Table
4 involving three groups and repeated measurements data.
13
15
Table 4
Repeated Measurements, Timm (1975, p. 454)
Subjects 1
Conditions
2 3
1 2 4 7
2 2 6 10
Drug group 1 3 3 7 10
4 7 9 11
5 6 9 12
Means 4 7 10
1 5 6 10
2 4 5 10
Drug group 2 3 7 8 11
4 8 9 11
5 11 12 13
Means 7 8 11
1 3 4 7
2 3 6 9
Drug group 3 3 4 7 9
4 8 8 10
5 7 10 10
Means 5 7 9
Grand mean 5.333 7.333 10
Using PROC GLM we test the standard MANOVA test for groups, conditions, and parallelism (group x
condition interaction) using Program 2.sas in Appendix B. Finding the differences in groups to be significant,
suppose we are interested in the extended linear hypothesis w = 0 where w is given (3.9). Using the PROC IML,
we verify the calculations of the overall test and estimate
yi = Tr(Ge) = Tr(GColike)= Tr(A0GC0i1)
= Tr(G*11)= 2.5
with the matrices G, G*, and B defined:
1 .5 1 .5 .5 4 7 10
G = .5[ .5 , Ga = .5 1 .5 and a = 7 8 11
.5 1 .5 .5 1 5 7 9
.
Solving IH 8 E;11= 0 with H = GW0G', Wo = Co (XX)-1C0 and Eo = A'oEiko =
X(X'X)-I X')YA0, we solve the characteristic equation using the IML routine EIGVAL for a symmetric
matrix. Because of rounding, the last characteristic root is essentially zero and, Sl = 6.881 and ;52 = 2.119. By
(3.4),
Trace = Tr(GW0G' Eo) = I
for the Bartlett-Lawley-Hotelling trace criterion. The corresponding value of o for largest root criterion is
&root = E ;"
as shown by Krishnaiah et al. (1980).
Since if = 23, the extended trace and root statistics for testing y/ = 0 are:
111/1/6-Trace = 0.8333
It//I /6,o°; = 1.3320.
Evaluating (2.13) with a = 0.05 for the trace criterion and a similar result for Roy's largest root test, the critical
values for the criteria are 1.331991 and 0.9891365, respectively. As shown in the program output, approximate
confidence intervals for 1,11 are (-53.77, 48.78) and (-6.50, -1.50) indicating nonsignificance of the contrast y/ for
both criteria.
We next illustrate testing (3.10) using the PROC IML by evaluating the supremum in (3.5) as outlined in (3.6).
With ri = Tr(Gie) and ty = Tr(GiWo Gi E0), we have in the Output that
r'= (-3, 1, 1, 2)
T=
( 29.6 26 13 7 \26 27.2 13.6 7.6
13 13.6 27.2 152
7 7.6 15.2 10.4
so that = 2.2081151. Comparing this with the U(G) critical value (1.3319921)2 = 1.774203, we see that the
test is significant. To find confidence intervals for contrasts involving y/i = Tr(Gie), one would find a 1 a
simultaneous confidence interval for each contrast as illustrated above. Finally, 95% confidence intervals are
constructed for yr in (3.11) following the test of parallelism, and for y/ in (3.12) following the test of equal
condition vectors, Program 2.sas.
5. An extended linear MMM hypothesis test
As the within subject design in a repeated measurements experiment becomes more complex, we often find that
the multivariate sphericity condition is not satisfied. For complex designs, the more general form of the hypothesis
in (2.3) becomes
H:e = CB(L ®A) = 0 (5.1)
15
17
where Lp,p is an orthogonal contrast matrix of rank p,L'L = Ip. Given (5.1), the covariance structure for (2.1) is:
= (L 0 A)'E(L ®A) (5.2)
and e = CB (L 0 A). Using the DMM, one may test (5.1) or (3.2) for arbitrary S2.
We can also test (5.1) or (3.2) given multivariate sphericity. Suppose, however, that 12 does not satisfy
the multivariate sphericity condition, but has general Kronecker structure:
= Ee eiu (5.3)
where Ee (p x p) and Eu = A'EA(u x u) are arbitrary positive definite matrices, a structure commonly found in
three-mode factor analysis models, Bentler and Lee (1978).
To test the null hypothesis that S2 has the structure given in (5.3) versus the alternative that the rows
y, of Y have a general structure, under multivariate normality, we obtain a likelihood ratio test using the normal
likelihood, Timm (1975, p. 558). The likelihood ratio statistic for testing the covariance structure is given by
i 1N/2= =
I A'±eAINPI2 I Eu IN(q-1) /2(5.4)
whereE, and Eu are the maximum likelihood (ML) estimates of Ee and Eu under the null hypothesis,
N NS = E (y, y)(y1 3)' / N is the ML estimate of the covariance matrice and y = E y1 / N.
1=1 1 =1
To test for the structure, the statistic -210, is asymptotically distributed as a chi-square distribution with
v degrees of freedom where v= pq(pq +1) 12 [p(p + 1) + q(q +1)+1)12 = (p 1)(q 1)[(p +1)(q +1)+1)12.
However, to solve the likelihood equations to obtain Ee and Eu involves an iterative process as outlined by
Krishnaiah and Lee (1980), Boik (1991),and Naik and Rao (1996). Naik and Rao provide a computer program
using the SAS IML procedure to obtain the ML estimates.
Given that 12 satisfies (5.3) and Ee and Eu are estimated by Ee and Eu , we may develop a test of the
extended linear hypothesis
H: = Tr(Ge) = 0.
Following Mudholkar et al. (1974), a test statistic for testing (5.5) is
X2 = {U(G)}2 = [Tr(G6)]2 / Tr[GC(X'X)-i C'G(Ee Eu
since
(5.5)
(5.6)
AlOy,&21if = Tr[GC(X'X)-i CG(Ee E)] (5.7)
for fixed G. The statistic X2 converges to a chi-square distribution with one degree of freedom.
The statistic in (5.7) is an alternative to Boik's adjusted multivariate test procedure for the more general
hypothesis given in (5.1) when multivariate sphericity is not satisfied. While we have provided a asymptotic test of
(5.1) given (5.3), the more difficult problem is the estimation of Ee and Eu, a solution to the likelihood in (5.4)
over all positive definite matrices Ee and Eu. Naik and Rao (1996) have developed an alternative Satterthwaite
type approximate for the MANOVA model when multivariate sphericity is not satisfied.
References
Amemiya, Y. (1994). On multivariate mixed model analysis. In T. W. Anderson, K. T. Fang, & I. Olkin (Eds.)Multivariate Analysis and Its Applications, IMS Lecture Notes, Volume 24, 83-95. IMS: Hayward, CA.
Bent ler, P. M. & S. Y. Lee (1978). Statistical analysis of a three-mode factor analysis model. Psychometrika, 43,343-352.
Boik, R. J. (1988). The mixed model for multivariate repeated measures: Validity conditions and an approximatetest. Psychometrika, 53, 469-486.
Boik, R. J. (1991). Scheffe's mixed model for multivariate repeated measures a relative efficiency evaluation.Communications in Statistics, A20, 1233-1255.
Boik, R. J. (1993). The analysis of two-factor interactions in fixed effects linear models. Journal of EducationalStatistics, 18, 1-40.
Bradu, D. & K. D. Gabriel (1974). Simultaneous statistical inference on interactions in two-way analysis ofvariance. Journal of the American Statistical Association, 69, 428-436.
Diggle, P.J., K.Y. Liang. and S.L. Zeger (1994). Analysis of longitudinal data. Oxford: Oxford University Press.
Graham, A. (1981). Kronecker products and matrix calculus: With applications, New York: Wiley.
Hecker, H. (1987). A generalization of the GMANOVA-model. Biometrical Journal, 29, 763-790.
Krishnaiah, P. R. & J. C. Lee (1980). Likelihood ratio tests for mean vectors and covariance matrices. In P. R .Krishnaih (Ed.) Handbook of Statistics, Analysis of Variance, Volume I, 513-570. New York: North-Holland.
Krishnaiah, P. R., G. S. Mudholkar, & P. Subbaiah (1980). Simultaneous test procedures for mean vectors andcovariance matrices. In P. R. Krishnaih (Ed.) Handbook of Statistics, Analysis of Variance, Volume I, 631-672. New York: North-Holland.
Lindsey, J. K. (1993). Models for repeated measurements. Oxford: Oxford University Press.
Longford, N. T. (1993). Random coefficient models. Oxford: Oxford University Press.
Mudholkar, G. S., M. L. Davidson & P. Subbaiah (1974). Extended linear hypotheses and simultaneous tests inmultivariate analysis of variance, Biometrika, 61, 467-477.
Muller, K. E., L.M. La Vange, S.L. Ramey, & E. T. Ramey (1992). Power calculations for general linearmultivariate models including repeated measures applications. Journal of the American Statistical
Association, 87, 1209-1226.
Naik, D. N. and S. S. Rao (1996). Analysis of multivariate repeated measurements (unpublished manuscript, Dept.of Mathematics and Statistics, Old Dominion University).
Pillai, K.C.S. (1960). Statistical Tables for Tests of Multivariate Hypotheses. Manila: Statistical Center,University of Philippines.
Pillai, K.C.S., and D.L. Young (1971). On the exact distribution fo Hotelling's generalized Toe, Journal ofMultivariate Analysis, 1, 90-107.
17
Potthoff, R.F. and S.N. Roy (1964). A generalized multivariate analysis of variance model useful especially forgrowth curve problems. Biometrika, 51, 313-326.
Reinsel, G. (1982). Multivariate repeated-measurement or growth curve models with multivariate random-effectscovariance structure. Journal of the American Statistical Association, M 190-195.
SAS (1990). SAS/STAT User's Guide, Version 6, (4th ed.), Vol. 1. Cary, NC: SAS Institute, Inc.
Seber, G.A. F. (1984). Multivariate Observations. New York: John Wiley.
Thomas, D. R. (1983). Univariate repeated measures techniques applied to multivariate data. Psychometrika, 48,451-464.
Timm, N. H. (1980) Multivariate analysis of variance of repeated measurements. In P. R. Krishnaiah (Ed.)Handbook of Statistics, Analysis of Variance, Volume I, 41-87. New York: North Holland.
Timm, N. H. (1975). Multivariate analysis with application in education and psychology. Monterey, CA: BrooksCole. [Reprinted by the Digital Printshop, 528 East Lorain Street., Oberlin, OH 44074 ISBN 0-7870-0008-6]
von Rosen, D. (1991). The growth curve model: A review. Communications in Statistics, 20, 2791-2822.
18 20
Appendix A
/* Program 1.sas *//* Example from Timm(1980) and Boik(1988,1991) Zullo Dental data */options LS=78 ps=60 nodate nonumber;title 'Output: Double Multivariate Linear Model and Multivariate Mixed Model';data dmlm;
infile 'mmm.dat';input group yl - y9;
proc print data =dmlm;proc glm;
class group;model yl - y9 = group/nouni;means group;
/* Multivariate test of group differences for mean vectors */manova h=group /printh printe;
proc glm;class group;model yl - y9 = group / nouni;
/* Multivariate test of Parallelism */contrast 'Parallel'
class group subj cond;model yl - y3 = group subj(group) cond cond*group;random subj(group);contrast 'Group' group 1 -1/e=subj(group);manova h = cond group*cond/printh printe;
/* Test for Multivariate Spericity and calculation of Epsilon for MMM */proc iml;print 'Test of Multivariate Sphericity Using Chi-Square and Adjusted Chi-Square Statistics;e=( 9.6944 7.3056 -6.7972 -4.4264 -0.6736 3.7255,
trace(s21)##2+trace(s21*s21)+trace(s22)##2+trace(s22*s22));epsilon=enum/eden;nu_h=nu_h#q; nu_e=nu_e#q; s0=min(nu_h,p);Mnu_h=nu_h#epsilon; Mnu_e=nu_e#epsilon; ms0=min(mnu_h,p);m0=(abs(mnu_h-p)-1)/2; n0=(mnu_e-p-1)/2;denom=s0##2#(2#m0+ms0+1); numer=2#(ms0#n0+1);dfl=ms0#(2#m0+ms0+1); df2=2#(ms0#n0+1);print 'Epsison adjusted F-Statistics for cond and group X cond MMM tests';print epsilon;f cond = df2#13.75139851/(ms0#dfl);f_gXc = df2#0.19070696/(ms0#df1);print f cond f_gXc dfl df2;p_cond=l-probf(f cond,dfl,df2);p_gXc=1-probf(f_gXc,dfl,df2);print 'Epsilon adjusted pvalues for MMM tests using TO**2 Criterion;print p_cond p_gXc;
21 23
Appendix B
* Program 2.sas *//* Data from Timm(1975, page 454) */options ls=78 ps = 60 nodate nonumber;title 'Output: Extended Linear Hypotheses' ;data timm;
infile 'exlin.dat;input group yl y2 y3 xl x2 x3;
proc print data=timm;proc glm;
class group;model yl-y3 = group/nouni;means group;
/* Multivariate test of Groups */manova h=group/printh printe;
proc iml;use Timm;a={xl x2 x3};b = {yl y2 y3};read all var a into x;read all var b into y;beta=inv(xs*x)*xs*y;print beta;n=nrow(y);p=ncol(y);k=ncol(x);nu_h=2; u=3; nu_e=n-k; s0=min(nu_h,u); r=max(nu_h,u); alpha=.05;denr=(nu_e-r+nu_h);roy_2=(r/denr)*finv(1-alpha,r,denr);rvalue=sqrt(roy_2);m0.(abs(nu_h-u)-1)12; n0=(nu_e-u-1)/2;num=s0**2*(2*m0+s0+1); dent=2*(s0 *n0+1);df=s0*(2*m0+s0+1);t0_2=(num/dent)*finv(1-alpha,df,dent);tovalue=sqrt(t0_2);print sO m0 n0;e=(y'*y-ys*x*beta);co={ 1 -1 0, 0 1 -1 };ao=i(3);eo=aos*e*ao;bo=co*beta*ao;wo=co*inv(x'*x)*cos;ho=bo'*inv(wo)*bo;print,"Overall Error Matrix", eo ,"Overall Hypothesis Test Matrix",ho;/* c'c=eo where c is upper triangle Cholesky matrix */c=root(eo);f=inv(cs) *ho*inv(c);eig=Eigval(round(f,.0001));vec=inv(c) *eigvec(round(C.0001));print,"Eigenvalues & Eigenvectors of Overall Test of Ho (Groups)", eig vec;/* Extended Linear Hypothesis following Overall Group Test */m={ 1 .5, -.5 .5, -.5 -1};g=ao*m*co;print, "Extended Linear Hypothesis Test Matrix",m g;psi=m*bo;
2224
psi_hat=trace(psi);tr_psi=abs(psi_hat);h=m*wo*ms;eo=inv (eo);c=root(eo);f= inv(c') *h *inv(c);xeig=Eigval(round(f,.0001));print, "Eigenvalues of Extended Linear Hypothesis", xeig;to_2=tr_psi/sqrt(sum(xeig)); print, "Extended To**2 Statistic", to_2;print, "Extended To**2 Critical Value", tovalue;root=tr_psi/sum(ssq(xeig)); print, "Extended Largest Root Statistic", root;print, "Extended Largest Root Critical Value", rvalue;print psi_hat alpha;ru=psi_hat+rvalue*sum(ssq(xeig));rl=psi_hat-rvalue*sum(ssq(xeig));vu=psi_hat+tovalue*sqrt(sum(xeig));vl=psi_hat-tovalue *sqrt(sum(xeig));print 'Approximate Simultaneous Confidence Intervals;print 'Contrast Significant if interval does not contain zero;print 'Extended Root interval: ('rl ru ')';print 'Extended Trace interval: ('vl vu ')';/* Multiple Extended Linear Hypothesis using To**2 */m1={ 1 0,0 0,0 0}; m2={0 0,1 0,0 0}; m3={0 0,0 1,0 0}; m4={0 0,0 0,0 1};print,"Multiple Extended Linear Hypothesis Test Matrices", ml,m2,m3,m4;gl=ao*ml*co; g2=ao*m2*co; g3=ao*m3*co; g4=ao*m4*co;tl=trace(m1*bo); t2=trace(m2*bo); t3=trace(m3*bo); t4=trace(m4*bo);tau=t1//t2//t3//t4;tl 1=trace(ml*wo*mr*eo);t21=trace(m2*wo*m1'*eo); t22=trace(m2*wo*m2s*eo);t31=trace(m3*wo*m1s*eo); t32=trace(m3*wo*m2'*eo); t33=trace(m3*wo*m3s*eo);t41=trace(m4*wo*m1'*eo); t42=trace(m4*wo*m2**eo); t43=trace(m4*wo*m3s*eo);t44=trace(m4*wo*m4s*eo);r1=t1111t2111t3111t41;
r2=t2111t2211t3211t42;
r3=t3111t3211t3311t43;
r4=t4111t4211t4311t44;
t=r1//r2//r3//r4;print tau,t;to_4=tau'*inv(t)*tau;print, "Extended Linear Hypothesis Criterion To**2 Squared", to_4;print, "Extended To**2 Critical Value", t0_2;/* Multivariate test of Parallelism */data timm;infile 'exlin.dat;input group yl y2 y3 xl x2 x3;
proc glm;class group;model yl-y3 = group/nouni;manova h = group m = ( 1 -1 0,
0 1 -1) prefix = diff/printe printh;proc iml;use Timm;a={x1 x2 x3 };b = {yl y2 y3};read all var a into x;
23 25
read all var b into y;beta=inv(xs*x)*xs*y;n=nrow(y);p=ncol(y);k=ncol(x);nu_h=2; u=2; nu_e=n-k; s0=min(nu_h,u); r=max(nu_h,u); alpha=.05;denr=(nu_e-r+nu_h);roy_2=(r/denr)*finv(1-alpha,r,denr);rvalue=sqrt(roy_2);m0=(abs(nu_h-u)-1)/2; n0=(nu_e-u-1)/2;num=s0**2*(2*m0+s0+1); dent=2*(sO*n0+1); df=s0*(2*m0+s0+1);t0_2=(nurn/dent) *finv(1-alpha,dfident);tovalue=sqrt(t0_2);print sO m0 n0;e=(ys*y-y'*x*beta);co =(1 -1 0, 0 1 -1);ao=(1 0, -1 1, 0 -1);eo=aos*e*ao;bo=co*beta *ao;wo=co*inv(xs*x)*cos;ho=bo'*inv(wo)*bo;c=root(eo);f= inv(c') *ho *inv(c);eig=eigval(round(f,.0001));vec=inv(c)*eigvec(round(f,.0001));print,"Eigenvalues & Eigenvectors of Overall test of Ho (Parallelism)", eig vec;/* Extended Linear Hypothesis following overall Parallelism test */m =(01,1 01;g=ao*m*co;print, "Extended Linear Hypothesis Test Matrix", m g;psi=m*bo;psi_hat=trace(psi);tr_psi=abs(psi_hat);h=m*wo*ms;eo=inv(eo);c=root(eo);f= inv(c') *h *inv(c);xeig=eigval(round(f,.0001));print, "Eigenvalues of Extended Linear Hypothesis", xeig;to_2=tr_psi/sqrt(sum(xeig)); print, "Extended To**2 Statistic", to_2;print,"Extended To**2 Critical Value", tovalue;root=tr_psi/sum(ssq(xeig)); print, "Extended Largest Root Statistic", root;print, "Extended Largest Root Critical Value", rvalue;print psi_hat alpha;ru=psi_hat+rvalue*sum(ssq(xeig));rl=psi_hat-rvalue*sum(ssq(xeig));vu=psi_hat+tovalue*sqrt(sum(xeig));vl=psi_hat-tovalue*sqrt(sum(xeig));print 'Approximate Simultaneous Confidence Intervals;print 'Contrast Significant if interval does not contain zero;print 'Extended Root Interval: ('rl ru ');print 'Extended Trace Interval: ('vl vu ') ;/* Multivariate test of Conditions as vectors */data timm;
infile 'exlin.dat';
2624
input group yl y2 y3 xl x2 x3;proc glm;
class group;model yl-y3 = group/noint nouni;contrast 'Mult Cond' group 1 0 0,
proc iml;use timm;a={xl x2 x3};b = {yl y2 y3);read all var a into x;read all var b into y;beta=inv(x'*x)*xs*y;n=nrow(y);p=ncol(y);k=ncol(x);nu_h=3; u=2; nu_e=n-k; s0=min(nu_h,u); r=max(nu_h,u); alpha=.05;denr=(nu_e-r+nu_h);roy_2=(r/denr)*finv(1-alpha,r,denr);rvalue=sqrt(roy_2);m0Mabs(nu_h-u)-1)/2; n0=(nu_e-u-1)/2;num=s0**2*(2*m0+s0+1); dent=2*(sO*n0+1); df=s0*(2*m0+s0+1);t0_2=(num/dent)*finv(1-alpha,df,dent);tovalue=sqrt(t0_2);print sO m0 n0;e=(y'*y-y'*x*beta);co=i(3);ao={1 0, -1 1,0-1);eo=aos*e*ao;bo=co*beta*ao;wc=co*inv(x'*x)*cos;ho=bo'*inv(wo)*bo;c=root(eo);f=inv(cs)*ho*inv(c);eig=eigval(round(f,.0001));vec=inv(c)*eigvec(round(f,.0001));print, "Eigenvalues & Eigenvectors of Overall test of Ho (Conditions)", eig vec;m=11 0 1, 0 1 1);g=ao*m*co;print, "Extended Linear Hypothesis Test Matrix", m g;psi=m*bo;psi_hat=trace(psi);tr_psi=abs(psi_hat);h=m*wo*m ;eo=inv(eo);c=root(eo);f= inv(c') *h *inv(c);xeig=eigval(round(f,.0001));print, "Eigenvalues of Extended Linear Hypothesis", xeig;to_2=tr_psi/sqrt(sum(xeig)); print, "Extended To**2 Statistic", to_2;print, "Extended TO**2 Critical Value", tovalue;root= trpsi/sum(ssq(xeig)); print, "Extended Largest Root Statistic", root;
print, "Extended Largest Root Critical Value", rvalue;print psi_hat alpha;ru=psi_hat+rvalue*sum(ssq(xeig));rl=psi_hat-rvalue *sum(ssq(xeig));vu=psi_hat+tovalue*sqrt(sum(xeig));vl=psi_hat-tovalue*sqrt(sum(xeig));print 'Approximate Simultaneous Confidence Intervals;print 'Contrast Significant if interval does not contain zero';print 'Extended Root Interval: ('rl ru ')';print 'Extended Trace Interval: ('vl vu ')';
I.
U.S. DEPARTMENT OF EDUCATIONOffice of Educational Research and Improvement (OERI)
Educational Resources Information Center (ERIC)
REPRODUCTION RELEASE(Specific Document)
DOCUMENT IDENTIFICATION:
Title:
Full Rank Multivariate Repeated Measurement Designs and Extended LinearHypotheses
Author(s):Neil H. Timm
Corporate Source:
University of Pittsburgh
Publication Date:
October 1, 1996
II. REPRODUCTION RELEASE:
Fri
In order to disseminate as widely as possible timely and significant materials of interest to the educational community, documentsannounced in the monthly abstract journal of the ERIC system, Resources in Education (RIE), are usually made available to usersin microficne, reproduced paper copy, and electronidoptical media, and sold through the ERIC Document Reproduction Service(EDRS) or other ERIC vendors. Credit is given to the source of each document, and, if reproduction release is granted, one of thefollowing notices is affixed to the docuMent.
If permission is granted to reproduce the identified document, please CHECK ONE of the following options and sign the releasebelow.
4. Sample sticker to be affixed to document Sample sticker to be affixed to document 11*
Check herePermittingmicrofiche(4" x 6" film),paper copy,electronic, andoptical mediareproduction.
"PERMISSION TO REPRODUCE THISMATERIAL HAS BEEN GRANTED BY
C7CN>
TO THE EDUCATIONAL RESOURCESINFORMATION CENTER (ERIC)*
Level 1
"PERMISSION TO REPRODUCE THISMATERIAL IN OTHER THAN PAPER
COPY HAS BEEN GRANTED BY
SaTO THE EDUCATIONAL RESOURCES
INFORMATION CENTER (ERIC)"
Level 2
or here
Permittingreproductionin other thanpaper copy.
Sign Here, PleaseDocuments will be processed as indicated provided reproduction quality permits. If permission to reproduce is granted, but
neither box is checked, documents will be processed at Level 1.
"I hereby grant to the Educational Resources Information Center (ERIC) nonexclusive permission to reproduce this document asindicated above. Reproduction from the ERIC microfiche or electronic /optical media by persons other than ERIC employees and itssystem contractors requires permission from the copyright holder. Exception is made for non-profit reproduction by libraries and other
service agencies atisfy information needs of educators in response to discrete inquiries."
Signature: .....e./y ----- Position:Professor
Printed Name:Neil H. Timm
Organization:University of Pittsburgh
Address:Dept. of Psychology in Education5C01 Forbes QuadranglePittsburgh, PA 15260